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Abstract
Reconstructive spectroscopy in the mid-infrared (MIR) range is an attractive method for miniaturizing MIR spectrometers. Although detectors with a sharp responsivity spectrum and a high dynamic range are required, there remains room for improvement. This article reports on a set of MIR photodetectors that satisfy these requirements with aluminum-covered narrow trench gratings. We experimentally found the optimal grating design, by which the responsivity changed 0.75% of the maximum value per nanometer of wavelength, and the minimum/maximum ratio was 8.7% in the wavelength range of 2.5–3.7 µm. A detector set of this optimal design was capable of accurate and robust reconstruction against measurement errors. This simple design structure based on standard materials will assist in realizing miniature MIR spectrometers.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Spectroscopy in the mid-infrared (MIR) region (2.5–25 µm in wavelength) has been exploited as one of the most efficient methodologies for determining chemical and biological properties [1]. In addition to the conventional benchtop spectrometers used in the laboratory, miniaturized spectrometers that provide on-the-spot results have recently drawn attention [2] for applications such as remote sensing [3,4], planetary exploration [5], gas detection [6–8] and food safety monitoring [9,10]. Although efforts have been made to miniaturize conventional spectrometers based on dispersive optics [11,12], Fourier transform [12–15] and narrow bandpass filters [16–18], a totally different framework of reconstructive spectroscopy [19–28] has emerged as one of the most promising approaches. Reconstructive spectroscopy is a numerical reconstruction of the incident spectrum using a set of detectors with uneven responsivity spectra that differ from one detector to another. Reconstructive spectroscopy has merit in that the spectral resolution is independent of the optical light path length [2], which is the main problem in minimizing the size of a spectrometer. Since the detectors will generate diverse responses according to their responsivity spectra and the spectrum of the incident light, the numerical reconstruction of the incidence spectrum is possible based on the premeasured responsivity spectra (Fig. 1(a)). An uneven responsivity spectrum is usually realized either by using an external optical filter before wavelength-independent MIR detectors [19–26] (called the “filter-array-detector-array (FADA)” configuration [29,30]) or by providing a wavelength-dependent function to the detector itself [27,28] (“filter-free” configuration [30]). The FADA configuration has a merit of feasible development assisted by the use of ready-to-use commercial MIR detectors, whereas the “filter-free” configuration has a potential for further miniaturization [2,30].
Fig. 1. Concept of this study. (a) Concept of reconstructive spectroscopy, (b) ideal responsivity spectrum with key factors for reconstructive spectroscopy, and (c) working principle of the proposed detector.
Despite intensive research efforts thus far on the side of numerical processing of reconstructive spectroscopy [28,31–34], which is the postprocessing of the measured data, strategies on the measurement itself have yet to be established, especially on how to realize responsivity spectra that are suitable for MIR reconstructive spectroscopy. The “suitable” responsivity spectra need to be capable of accurate reconstruction of the true incident spectrum with robustness against measurement error. Our interest is to establish a strategy to realize the responsivity spectrum suitable for “filter-free” reconstructive spectroscopy. This article explains a design of a set of thermal MIR detectors that realize responsivity spectra suitable for reconstructive spectroscopy in the MIR.
Robust reconstruction against measurement error requires a sufficient difference between one responsivity spectrum and another. Similar spectra close to each other will have only a slight difference in output that would be strongly affected or overwhelmed by measurement errors, resulting in serious reconstruction errors. From a mathematical point of view, the difference in the responsivity spectra corresponds to the linear independence of the “basis vectors” used for the reconstruction. A good design of a reconstructive spectrometer requires a strategy to secure linear independence in an engineerable manner. In this regard, shifting a responsivity spectrum with a distinctive feature is a preferable approach.
The concept of shifting the responsivity spectrum has been employed implicitly in reconstructive spectroscopy [20–27,35–37]. Features of the responsivity spectra appearing at different wavelengths ensure linear independence. In so doing, sharpness of the feature and a high dynamic range are important. A spectrum consisting only of gradual slopes will result in a signal hardly distinguishable from that of a shifted one. Likewise, a large offset of responsivity will degrade the distinction between the devices. The sharpness, dynamic range and capability of shifting the wavelength in an engineerable manner are the main factors for securing linear independence (Fig. 1(b)).
In this article, we report on thermal MIR detectors with a one-dimensional plasmonic grating consisting of aluminum-coated narrow trenches, as shown in Fig. 1(c). The detectors exhibit a responsivity spectrum with a sharp feature and a wide dynamic range, functioning as components of the “filter-free” reconstructive spectrometer. This study was motivated by our previous findings of the spectral response of a narrow trench grating that might be applicable to reconstructive spectroscopy [38]. We experimentally examined the device response to MIR light and obtained a trench design that optimizes the sharpness and the dynamic range. Based on the optimized design, the shifting of the responsivity spectrum was possible without deteriorating the sharpness and the dynamic range. Using the measured responsivities from twenty grating structures, we numerically demonstrated the reconstruction robustness against measurement noise without degrading accuracy, and the spectral resolution of the spectrometer was also obtained. With its robust structure without any moving parts or external filters and simple fabrication process consistent with conventional semiconductor processes, our detector will contribute to the realization of a practical miniaturized spectrometer in the mid-infrared range.
2. Materials and methods
2.1 Device design and fabrication
The periodic structure of the grating consists of a narrow trench, the width of which is defined with respect to its periodicity, or the “pitch”, continuously covered by aluminum. The grating structure can be defined by the grating pitch (p), trench width (w) and depth (d) and thickness of the aluminum layer (t). When MIR light is incident upon the grating, the light is absorbed through the generation of surface plasmons that make the device response dependent on wavelength and the grating structure. The absorbed photon energy is then converted to heat. The structure- and wavelength-dependent MIR signals can thus be obtained by measuring temperature-dependent physical properties such as resistivity. Therefore, it is essential to reflect as much incident energy as possible at the least responsive wavelength while keeping the efficient light absorption at the resonance condition, to achieve a high dynamic range in the responsivity spectrum. In this sense, the continuous aluminum layer should be the key component in the structure, working as a partition that prevents direct penetration of light into the silicon substrate where light is absorbed via the free carrier absorption (FCA).
To characterize the role of the grating parameters in the responsivity spectra, we designed and fabricated thermal MIR detectors, as illustrated in Fig. 2(a). The grating was placed between two gold electrodes, between which a constant bias voltage was applied. The gold was in ohmic contact with p-type silicon. The current between the electrodes varied as the resistivity of the silicon substrate changed according to the temperature.
Fig. 2. Design and fabrication of the device. (a) Device design with grating parameters, (b) schematic of the fabrication process and (c) SEM photographs of the cross section of the grating.
The device was fabricated by electron-beam (EB) lithography and inductively coupled plasma-reactive ion etching (ICP-RIE), known as the Bosch process, followed by the deposition of aluminum that covered the grating and the formation of the gold electrodes (Fig. 2(b)). A boron-doped p-type silicon substrate with a resistivity of 0.17 Ω-cm was used. In general, the Bosch process is known to yield ripples called scalloping on the sidewall, arising from the reciprocal exposures to passivation and etching. However, a rough surface is not preferable for a continuous metal layer that covers the whole surface of the grating. We avoided the formation of these ripples by shortening the period of each process to as short as 1 second and increasing the number of cycles. In the deposition of aluminum on the grating, we first removed the nature oxide layer by fluoric acid and placed the chip on a rotary stage in the vacuum chamber with a 20-degree tilt so that the trench was coated by a continuous metal layer. The micrograph obtained by scanning electron microscopy (SEM) in Fig. 2(c) shows the cross section of the grating. The device was designed with a w = 297 nm, d = 500 nm, and t = 50 nm. The deposited aluminum is thicker on the surface of the substrate than on the sidewall trench by a factor of approximately 1/tan20° (≈2.7). Note that a contact of aluminum and p-type silicon forms a rectifying Schottky junction, which prevents a current through the substrate from short-cut via aluminum layer, as opposed to the ohmic gold/p-silicon contact [38]. After etching aluminum deposited out of the grating area using a photoresist mask, another photoresist pattern was made for lifting off the gold electrodes. The photoresist also protects the aluminum layer underneath from an exposure to fluoric acid that removes the nature oxide layer before the evaporative deposition of gold directly onto the silicon substrate. The fabricated device was glued onto a printed circuit board and wire-bonded.
2.2 Experimental
The device was irradiated by a wavelength-tunable MIR laser (Firefly IR-LP-A) through a long wavelength pass filter (cutoff wavelength 2.4 µm) that eliminated the excitation light of the laser (Fig. 3). The incident light was polarized perpendicular to the grating. The measuring wavelength was set to every 10 nm from 2.5 to 3.7 µm. The laser irradiated the device with a normal incident for 10 seconds followed by 30 seconds of cooling without irradiation. Between the two gold electrodes, a 0.1 V bias voltage was applied with a source meter (Keithley, 2614B), and the current was measured. The incident light power and the reflection on the device were also monitored using a beam splitter during the measurement. The responsivity at each wavelength was calculated as follows: First, the fractional change in resistance (ΔR/R) was calculated from the measured voltage and current at each sampling point. The data collected during irradiation and cooling, which have a time constant that does not reach saturation within the measured time periods, were fitted with exponential curves with the same time constant to obtain the saturated values. Then, the difference between the saturated values of irradiation and cooling was divided by the incident power to obtain the responsivity. Reiterating the steps above at each wavelength yields a responsivity spectrum. The reflection, on the other hand, was divided by the incident power to give normal reflectance as a sign of the light-to-heat conversion. A direct measure for the conversion is absorption, which equals to the remainder of 1 subtracted by reflectance and transmittance. Still, when the transmittance is very small, the normal reflection gives a qualitative indication of the energy conversion process.
Fig. 3. Experimental setup for the responsivity characterization. The laser output contains excitation light with shorter wavelengths, which was eliminated by a long-wavelength pass filter with a cutoff wavelength of 2.4 µm. The readout circuit of the device was connected to a source meter that applied a bias voltage of 0.1 V.
As an indicator of the sharpness of the obtained responsivity spectra, we normalized the responsivity spectra by setting the minimum value to 1 and calculated the maximum gradient. The dynamic range was evaluated by the local minimum/maximum ratio across the maximum gradient slope as well as the global minimum/maximum ratio within the measured wavelength range.
3. Experimental results
3.1 Trench dimensions
We first examined the parameters concerning the periodic structure, namely, the aluminum thickness t, trench width w, and trench depth d, to experimentally obtain the optimal parameter set. The search parameters were chosen after the preliminary result [38].
Figure 4(a) shows the results with the aluminum thickness t as a variable (25, 50, 100 nm). The other dimensions are p = 2.97 µm, w = 297 nm, and d = 500 nm. For all the aluminum thickness conditions, the spectrum had a dip (denoted as “FD” in the following) at a wavelength that was approximately equal to the grating pitch (2.97 µm) and a peak (“FP”) at a longer wavelength by approximately 100 nm. Between the dip and the peak was the steepest slope in the spectrum. Another dip is seen around λ = 3.4 µm, followed by a more gradual slope. The reflectance varied correspondingly to the shape the responsivity spectra, indicating conversion of photon energy into thermal energy was done on the grating. The difference due to the thickness was a higher offset of responsivity spectrum for t = 25 nm than for the thicker conditions. The reflectance spectra had similar shapes for all the conditions, but that of the 25-nm-thick aluminum layer was slightly lower overall. As a result, t = 25 nm had a lower maximum gradient and a lower dynamic range than the remaining two conditions, which showed almost identical characteristics. This result might reflect the fact that an aluminum layer as thin as 25 nm partly transmits incident light, resulting in larger contribution from FCA in the doped silicon substrate underneath. On the other hand, the saturated maximum gradient and dynamic range for t = 50 nm and 100 nm suggest that 50 nm-thick aluminum layer was enough to suppress the transmission. Later in Section 5.1, we discuss the transmittance through the grating based on numerical simulation.
Fig. 4. Measured responsivity and reflectance spectra and analysis of the responsivity spectra. The variables are the (a) thickness of the aluminum layer t, (b) width of the trench w, and (c) depth of the trench d. The basic grating parameters are grating pitch p = 2.97 µm, w/p = 10%, d = 500 nm, and t = 50 nm. For each variable, the measured responsivity (i) and the reflectance (ii) is shown followed by (iii) the sharpness and dynamic range extracted from the responsivity spectra and (iv) the transition of feature points. In (iii), local minimum-to-maximum ratio across the largest gradient (black solid line), global minimum-to-maximum ratio (black dashed line), and largest gradient of normalized responsivity (minimum responsivity is set to 1) are plotted.
When the trench width w was a variable, the result shown in Fig. 4(b) was obtained. The other parameters were designed such that p = 2.97 µm, d = 500 nm, and t = 50 nm. The w/p ratio was set to 5.72 to 25.7%. As the width widens with respect to the grating pitch up to 15.7%, the peak FP around λ = 3.1 µm in the responsivity spectrum monotonically shifted to the shorter side of the wavelength, whereas the transition of the peak height was not monotonic. All the spectra reached the dip FD at approximately λ = 2.97 µm, identical to the grating pitch. When the w/p ratio was as large as 25.7%, the peak FP at approximately λ = 3.1 µm was no longer observed, and both the responsivity and reflectance deviated from the tendency of the other conditions. Among the varied width conditions, the w/p ratio of 12.5% had the largest maximum gradient (86 µm−1) and the highest dynamic range (global and local min/max ratio was 8.7% and 16%, respectively). With respect to the largest responsivity observed at λ = 3.07 µm, the maximum gradient amounts to 0.75% per nanometer of wavelength.
In Fig. 4(c), we show the result of the trench depth d as a variable with the other parameters being p = 2.97 µm, w/p = 10%, and t = 50 nm. Each responsivity spectrum is normalized by its minimum value. Tracing FP’s in the plot, the wavelength monotonously red-shifted as the trench depth d increased, but the responsivity peaked out between d = 484 nm and d = 607 nm (Fig. 4(c-i)). As a result, both the dynamic range and the maximum gradient of the responsivity spectrum were highest at d = 484 nm.
We determined the optimal parameters for a periodic structure from these measurements to be t = 50 nm, w/p = 12.5%, and d ≈ 500 nm.
3.2 Grating pitch as a spectrum-shifter
The remaining parameter, the grating pitch p, was found to have the capability to shift the responsivity spectrum while maintaining its sharpness and dynamic range. Figure 5(a) shows the normalized responsivities of 20 devices with different values of grating pitch p. These devices have the near-optimal design with t = 50 nm, d = 500 nm, and a constant w/p ratio of 10%. The grating pitch was set to every 60 nm from 2.43 µm to 3.57 µm. The trench depth was 500 nm, and the aluminum thickness was 50 nm. The result of the measurement showed that the position of the steepest slope shifted as the grating pitch lengthened. The lowest responsivity was observed at the dip FD (marked by red triangles) for each condition, except for p = 2.43 µm, where FD was out of the measurement range. Note that in the plot, each spectrum for p ≥ 2.49 µm was normalized so that the lowest responsivity equals 1, and the spectrum for p = 2.43 µm was adjusted so that it follows the trends of the other spectra. The aligned peaks FP (marked by black triangles) with almost the same height indicate that the dynamic range is not affected by the spectrum shift. To extract the wavelengths of FP and FD from the responsivity spectra smoothed by a Gaussian kernel and plotted against p, Fig. 5(b) was obtained. Both of the feature points FD and FP were aligned with almost the same gradient. Because of the unchanging dip-to-peak distance, the steepness remained almost constant for all pitch conditions, rather than getting flattened in proportion to the wavelength. This constant sharpness, regardless of the wavelength, makes this narrow trench grating suitable for fine resolution especially in longer wavelength ranges.
Fig. 5. Responsivity spectra obtained by the grating pitch variation. (a) Measured responsivity spectra with 20 different grating pitches, normalized so that the minimum becomes 1. (b) Locations of the peak and dip across the steepest slope.
4. Numerical estimation of reconstruction ability
4.1 Numerical method
We simulated the reconstruction of the incident spectrum using the experimentally obtained responsivity spectra shown in Fig. 5(a) by taking the steps as follows. First, we prepared a virtual spectrum that is to be incident upon the array of detectors. The detectors should respond to the incidence according to the following equation
In addition to the relationship between the incident spectrum and the detector response as shown in Eq. (1), real measurements contain measurement errors ɛ. The measured value Ireal should therefore be written as Ireal = RPtrue + ɛ. This equation simulates the detector responses Ireal to a certain incident spectrum Ptrue. The reconstruction is an inverse problem that solves for P based on the measured value I and the precalibrated responsivity matrix R. The simplest solution can be obtained by using the Moore-Penrose generalized inverse of the responsivity matrix R−1 as Pest = R−1Ireal.
We assume that each component of ${\boldsymbol \varepsilon }$ is a random number obeying the normal distribution
which has a standard deviation proportional to the mean sensor reads, i.e.,Here, σr is the noise level parameter that defines the signal-to-noise ratio. We will refer to this parameter σr simply as the “noise level” in the following sections. This assumption of the measurement error is based on an environment where the measurement error originates mainly from slow perturbation factors, such as unstable incident power or moving background black bodies, rather than the shot noise that comes from the current flowing through the detectors. We assume that the shot noise here is relieved by smoothing a number of data points or limiting the sampling bandwidth. In fact, to calculate the noise equivalent power (NEP) of the detector from the noise under no light irradiation, it was 1.00 µW/Hz1/2 (“t = 50 nm” device in Fig. 4(a), at 3.70 µm, integration time = 0.06 sec), while the laser incidence was more than a few milliwatts with a fluctuation of more than 1% during the measurement.
The reconstruction should be as close to the original spectrum as possible. We employed the normalized estimation error norm
4.2 Reconstruction result
To see the basic reconstruction performance of the measured responsivity matrix, we first employed Gaussian spectra as incidence as shown in Fig. 6(a). The spectrum is represented by
Fig. 6. Numerical reconstruction based on measured responsivity spectra. (a) Gaussian spectra used for evaluation of reconstruction accuracy and (b) result of the evaluation. (c) Reconstruction result for single-wavelength incidence. λ = 2.8, 3.1, 3.4 µm are shown. (d) Reconstruction result for two single-wavelength incidences separated by 80 nm. (e) Reconstruction of transmission spectra of glucose and sucrose. Original spectra are also shown by dashed lines.
We evaluated the wavelength resolution using a single wavelength spectrum that has a nonzero value at one component (incident wavelength) and zero for the others. The noise level is neglected here (i.e., σr = 0). Figure 6(c) is the reconstruction result for a single wavelength incidence with λ = 2.8, 3.1, 3.4 µm, which are indicated in the figure by dashed lines. A single peak was reconstructed at the incident wavelength. The wavelength resolution as full-width at half-maximum (FWHM) was 68 nm on average of the incident wavelength from 2.6 to 3.6 µm. The FWHM was large (i.e., poor wavelength resolution) when the incident wavelength fell on one of the FP’s and small in-between. This periodic nature of the resolution will be discussed later in the Discussion section. When two single wavelengths were simultaneously incident with the same intensity, a separation of 80 nm was distinguished as shown in Fig. 6(d). These obtained values (68 nm and 80 nm) are smaller than those predicted by Shannon-Nyquist sampling theory, which predicts a wavelength resolution of 120 nm when the sampling of an interval of 1.2 µm is performed by 20 measurements.
To demonstrate the practicality of reconstruction, we employed the transmission spectra of glucose and sucrose. The original spectra obtained from the database [39,40] were reconstructed with a noise level of 0.1%. We show the result in Fig. 6(e). Although some unwanted ripples are seen on the periphery of the spectral range, the two spectra are clearly distinguishable by the reproduced spectral features. The normalized estimation norm $\hat{E}$ was 9.7% for glucose and 19% for sucrose. The artifacts can be mitigated by the choice of numerical method, while some considerations are needed as discussed later in Section 5.2.
4.3 Evaluation of sharpness and dynamic range of responsivity spectrum
In this section, we discuss the role of the sharpness and the dynamic range of the responsivity spectra, which we qualitatively examined in the Introduction. To parameterize the sharpness and the dynamic range, we numerically deformed the responsivity we obtained in Section 3.2 and used it for reconstruction in the last section: expanding along the wavelength axis to vary the sharpness and inserting offset to compress the dynamic range (Fig. 7(a)). In expanding the spectra, the longer wavelength end (λ = 3.7 µm) was pinned and the rest were expanded to the shorter side before resampling by interpolation. Offsetting the spectrum was followed by rescaling by the largest responsivity value across the detectors. The sharpness was parameterized by the expansion factor, and the dynamic range was determined by the proportion of the offset to the maximum responsivity.
Fig. 7. Numerical reconstruction of Gaussian curves using degraded responsivity matrices. (a) Responsivity matrix deformed by expansion and offsetting. (b) Reconstruction performance of expanded responsivity matrices and (c) offset responsivity matrices. Dashed lines with square markers are the error with zero noise level for reference.
As we described in the Introduction, the sharpness and dynamic range are important factors for reconstruction against noise. However, the reconstruction must have the ability to replicate the spectrum accurately before discussing the robustness. To evaluate the reconstruction accuracy and the robustness against noise separately, measure the quality of the reconstruction, we define inaccuracy IA and noise sensitivity NS as follows:
Figure 7(b) shows the results for the expansion. Dashed lines with square markers indicate a zero-noise reference where IA was calculated with σr = 0. The IA monotonically increased in accordance with the expansion factor, except for the spectrum widths σin of 50 and 200 nm, for which the error bottomed when the responsivity was expanded by 1.5 times. In contrast, the zero-noise reference, an ideal condition, was lower even when the responsivity spectra were expanded. The noise sensitivity NS drastically increased according to the expansion factor. Indeed, an expansion by 1.5 times resulted in an NS 6.2 times larger than the original responsivity (on average over spectral width). These results indicate the noise vulnerability of the reconstruction when the sharpness of the responsivity spectra was degraded.
Likewise, the result for offsetting in Fig. 7(c) shows that the offset to the responsivity spectra can damage the robustness. IA was almost constant until the offset was up to 50%, meaning the offset did not affect the reconstruction when the measurement was accurate (i.e., low measurement noise). However, NS exponentially increased as the offset ratio increased. Although the dynamic range might have less impact than the sharpness, it is still an important factor of a good responsivity spectrum.
4.4 Evaluation of shifting width
In the measurement-based responsivity matrix, we fabricated a set of detectors whose grating pitch (and accordingly the responsivity spectrum) was shifted by 60 nm per detector. To evaluate the impact of shifting widths, we calculated intermediate pitch conditions as follows. First, the measured responsivity spectra were blue-shifted by their grating pitch (i.e., the measured responsivity was re-plotted with respect to λ [µm] – p [µm]) and normalized by the minimum value. After this procedure, the spectral features were aligned because the wavelength at which the steepest slope appears in the responsivity spectrum redshifts at the same rate as the increase of p. And then we linearly interpolated the data set (p, λ, responsivity). Data at the edge were linearly extrapolated when it was necessary. To discuss based on the measured responsivity, the shifting width was set to 60 nm or smaller.
To parameterize the shifting width, we first fixed the number of detectors and changed the shifting width (Fig. 8(a) left). Secondly, the spectral width was changed by increasing the number of detectors (Fig. 8(a) right). In the former condition, the steepest slopes of the detectors cover less spectral range, while in the latter the coverage wavelength range is fixed to be constant. The results are as shown in Figs. 8(b) and (c). When the number of detectors were fixed, both IA and NS monotonously increased as the shifting width was narrowed (Fig. 8(b)). For a fixed coverage, IA reached the bottom at the shifting width of 30 nm and maintained the same value for larger shifting widths up to 60 nm (Fig. 8(c)). However, NS became exponentially large as the shifting width decreased. These two results indicate that a shifting width of 60 nm was the best among the tested, to attain a good reconstruction quality and a robustness against noise. Note here that 60 nm is what we get by dividing the spectral range (1200 nm) by the number of detectors (20), and it is comparable with the spectral width of the steepest slope that spans approximately 100 nm. It is important to minimize the similarities between the responsivity spectra for a robust reconstruction against measurement noise, and it is also important to maintain the reconstruction accuracy by a sufficient number of available data.
Fig. 8. Reconstruction of Gaussian curves using different shifting widths of responsivity spectrum. (a) The interpolated responsivity spectra. (b) Reconstruction performance of a fixed number of detectors and (c) of a fixed coverage by the steepest slopes in the spectral range. For plot legends, see Fig. 7.
5. Discussion
5.1 Phenomenon behind the structure-dependent responsivity spectrum
Figure 9(a) shows the measured responsivity and reflectance spectra for a varied incident angle within a plane perpendicular to the grating. Although a normal incident (0 deg.) has one peak at approximately 3.1 µm and one dip at approximately 3.0 µm, another peak emerges from the dip as the incident angle deviates from 0 deg. The new peaks appear close to the theoretical SPR conditions given by [41,42]
Fig. 9. (a) Device response to a variation of incident angle and (b) a schematic of the possible response mechanism of the grating.
Fig. 10. Simulated photoresponse of the grating for oblique incidence. (a) Reflectance, (b) transmittance, and (c) electric field corresponding to the even and odd modes. The angle of incidence θ was set to every 1 deg from 0 to 5 deg, and the electric field for θ = 2 deg is shown.
The more energetically stable the odd mode is, the longer wavelength the peak appears at and the less steep the responsivity spectra become. Therefore, the steepness of the responsivity should be engineered by the shape of the trench. In the odd mode, charges with opposite parities face across the trench. The oscillating energy will be stabilized when the charges face closer to each other on a larger area. In fact, as shown in Fig. 4, the distance in wavelength from dip to peak depends on the dimensions of the trench: narrower and deeper trenches result in the furthering of this distance.
5.2 Reconstruction method
There are many methods to solve inverse problems, including annealing [33], Tikhonov regularization [23,27,31,36,44] and LASSO regularization [27,28,32,45,46], with and without nonzero constraints, respectively. However, these methods share the basic principle with the generalized inverse matrix: that the physics is described as a linear model I = RPtrue and that the residue ||RPest − I|| should be as small as possible. They improve the reconstruction performance by assuming more a priori information about the incident spectrum, such as the minimum base components [28,32,36,46] or the smoothness of the spectrum [31,44], which is sometimes unrealistic for natural spectra for which those spectrometers will eventually be used. For example, absorption by the C-H bond appears sharply at approximately 3.4 µm with more than one peak if multiple configurations reside in the molecule, whereas O-H bond absorption appears as a broad dip at approximately 2.9 µm. If these bonds coincide at the same time, neither the smoothness nor the minimum base component assumptions are applicable. It is essential that the detectors possess the ability to reconstruct without a priori information.
5.3 Practical device design
As we mentioned in the Introduction, linear independency among the responsivity spectra is important for the reconstruction to remain robust against measurement errors. We employed the concept of shifting the spectrum in this study to achieve linear independence. Another approach is to use random-like spectra. Motivated by the low correlation between random numbers, a set of spectra that fluctuate rapidly within the spectral range of interest were realized by multilayer filters in the visible region [46] or photonic crystals in the near-infrared region [44,47]. Although this approach is mathematically feasible, engineering capability becomes a problem. Optimization to a required specification is necessary for applications in practice. When the number of the required detectors or the spectral range are changed, it is almost impossible to design additional detectors and ensure the independence of the random-like spectra that appear to be a set of linearly independent bases. On the other hand, shifting a spectrum can be achieved, and its spectral shape can be easily adjusted by a parameterized device structure. Therefore, shifting a spectrum with distinctive features has merit from the viewpoint of engineering capability.
To step further into designing a responsivity matrix by shifting a spectrum, the wavelength span of the slope should be chosen based on the number of detectors used, the wavelength range of interest, the wavelength resolution tolerance of reconstruction and the measurement error under use. As we indicated in Section 4.2, the wavelength resolution is degraded around the wavelength of a spectral peak. When the incident wavelength corresponds to a peak of a spectrum where its gradient is zero and other spectra have relatively low values, the response from the detectors does not change as sharply as when the wavelength is between the peaks. The fluctuation of wavelength resolution is therefore inevitable to some extent but can be relieved by positioning the peaks closer to each other with respect to the dip-to-peak distance. A spectrum with a wide dip-to-peak distance or closely aligning the spectra will ease the fluctuation. However, the wide dip-to-peak distance realized by narrow and deep trenches, for example, should become less robust against noise. Likewise, closely aligned spectra will require more detectors to cover the same wavelength range as we have seen in Section 4.4. The dimensions of the trench should thus be carefully chosen from the requirements of application.
Note that the detector set used in the reconstruction has the same grating depth. This means the detector set is fabricated using a common recipe, allowing integration of the detectors into one chip. Besides, the grating functions as a light absorber that generates heat and the silicon substrate can be replaced by other material with a temperature-dependent resistivity. Therefore, the grating can be implemented into existing thermal infrared detectors to compose a miniature MIR spectrometer. Since the common thermal infrared detectors use materials of high temperature coefficient of resistivity such as vanadium oxide or amorphous silicon, an improved responsivity as well as higher signal-to-noise ratio can be expected from this implementation. Considering the successful miniaturization of thermal infrared detectors, further downsizing of MIR detectors as a component of a reconstructive spectrometer is a feasible approach.
Thermal MIR detectors have, by their working principle, no wavelength detection limit [48]. Since the proposed narrow trench presented a constant sharpness and a high dynamic range regardless of the grating pitch, we can expect the applicability of the proposed grating structure to extend to even longer wavelength ranges. From 7 to 20 µm is a range called the “fingerprint region” [49,50], where the absorption spectrum is unique due to the chemicals contained in the subject material. By widening the grating pitch to the target wavelength and conserving the w/p ratio to approximately 10%, we can expect a compact and rigid spectrometer that detects the “fingerprint” information.
6. Conclusion
We proposed a silicon-based thermal MIR detector that has a grating consisting of a narrow trench continuously covered by aluminum. Governed by the coupling condition of surface plasmon resonance, its responsivity spectrum shows a sharp feature with a high dynamic range, which is ideal for reconstructive spectroscopy. The optimized trench has an aluminum thickness of 50 nm, a trench width of 12.5% of the grating pitch, and a depth of approximately 500 nm. By varying the grating pitch, the responsivity spectrum can be shifted with a precise adjustability, maintaining its sharp features. Using 20 pitch conditions of the near-optimal structure and the most basic reconstruction algorithm of a generalized inverse matrix, a robust reconstruction against noise is possible with a wavelength resolution of 68 nm. Taking the fundamental result in this study that proves the applicability of the narrow trench plasmonic grating to the reconstructive spectroscopy, we can now get on to the integration of the detectors. The feasibility of the detector with a simple structure, the use of nontoxic and basic materials and the thermal response framework can assist the realization of portable and inexpensive miniature MIR spectrometers for versatile applications.
Funding
New Energy and Industrial Technology Development Organization.
Acknowledgments
Prof. Shoji Takeuchi is acknowledged for their fruitful discussion. EB lithography, ICP-RIE and dicing were performed in Takeda Clean Room operated by System Design Lab (d.lab), School of Engineering, the University of Tokyo, Japan, and other processes were performed in a clean room at the Division of Advanced Research Facilities of the Coordinated Center for UEC Research Facilities of the University of Electro-Communications (UEC-Darf), Tokyo, Japan. Infrared transmission spectra were obtained from SDBSWeb: https://sdbs.db.aist.go.jp (National Institute of Advanced Industrial Science and Technology, retrieved May 1st, 2022). This research was supported by the New Energy and Industrial Technology Development Organization (NEDO), Japan.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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